The prefix computation problem is to compute all n initial products a1 ∘ … ∘ ai, i = 1, …, n, of a set of n elements, where ∘ is an associative operation. We present an 0(((log n)/log(2n/p)) · (n/p)) time deterministic parallel algorithm using p ≤ n processors to solve the prefix computation problem, when the order of the elements is specified by a linked list. For p ≤ 0(n1-∊)(∊ > 0 any constant), this algorithm achieves linear speedup. Such optimal speedup was previously achieved only by probabilistic algorithms. Our algorithms assume the weakest PRAM model, where shared memory locations can only be exclusively read or written (the EREW model).